Quadratic Equations in Square Roots

Quadratic Equations in Square Roots

For solving Quadratic Equations in Square Roots, it is required to square both sides entirely, but not individually. For instance,
$$ \begin{align} (1 + 2)^2 &= 3^2 \\
1^2 + 2^2 &\ne 3^2 \end{align} $$
It is important to check the solutions to see if they work for the original equation if the original equation is squared.

Worked on Examples of Quadratic Equations in Square Roots

Solve \( \sqrt{x+5} + \sqrt{x-2} = \sqrt{5x-6} \) .

\( \begin{aligned} \require{color} \require{AMSsymbols} \displaystyle
\Big(\sqrt{x+5} + \sqrt{x-2}\Big)^2 &= \sqrt{5x-6}^2 &\color{red} \text{square both sides} \\
x+5 + 2 \sqrt{x+5} \sqrt{x-2} + x-2 &= 5x-6 \\
2 \sqrt{x+5} \sqrt{x-2} &= 3x-9 \\
4(x+5)(x-2) &= (3x-9)^2 &\color{red} \text{square both sides} \\
4(x^2 + 3x-10) &= 9x^2-54x + 81 \\
5x^2-66x + 121 &= 0 \\
(5x-11)(x-11) &=0 \\
x = \frac{11}{5} \text{ or } 11
\end{aligned} \)
\text{Check whether the answers are OK, but } x= \displaystyle \frac{11}{5} \text{ does not make the original equation.} \\
\text{So the answer is only } x=11.

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